<p>Let \(E - ABCD\) be a pyramid on square base \(ABCD\) where \(A\) is the origin and \(B\) and \(D\) are lying on positive \(x\)-axis and \(y\)-axis respectively. If \(E\) is \((0, 2, 3)\) and \(\overrightarrow{DE} \cdot (\hat{i} + \hat{j}) = 0\), then:</p>
<p>(a) image of the point \(D\) in the plane \(ABE\) is \(\left(0,\, \dfrac{-10}{13},\, \dfrac{24}{13}\right)\)</p>
<p>(b) image of the point \(D\) in the plane \(ABE\) is \(\left(0,\, \dfrac{-6}{13},\, \dfrac{30}{13}\right)\)</p>
<p>(c) volume of the tetrahedron \(ABDE\) is 2 cubic units</p>
<p>(d) perpendicular distance of the point \(D\) from the plane \(ABE\) is \(\dfrac{9}{\sqrt{13}}\)</p>
Step-by-Step Solution
Key Concept: Use the perpendicularity condition $\overrightarrow{DE} \cdot (\hat{i} + \hat{j}) = 0$ to find the side length of the square base, then leverage the coordinate system where $A$ is origin, $B$ on x-axis, and $D$ on y-axis.
Step 1: Set up coordinates. Since $A$ is origin, $B$ on positive x-axis, and $D$ on positive y-axis for square $ABCD$ with side length $a$: $A = (0,0,0)$, $B = (a,0,0)$, $D = (0,a,0)$, $C = (a,a,0)$, and $E = (0,2,3)$ (given). Step 2: Apply perpendicularity condition. $\overrightarrow{DE} = E - D = (0,2,3) - (0,a,0) = (0,2-a,3)$. The condition $\overrightarrow{DE} \cdot (\hat{i} + \hat{j}) = 0$ gives: $(0,2-a,3) \cdot (1,1,0) = 0 + (2-a) + 0 = 0$, so $a = 2$. Step 3: Determine vertex coordinates. With $a = 2$: $A = (0,0,0)$, $B = (2,0,0)$, $C = (2,2,0)$, $D = (0,2,0)$, $E = (0,2,3)$. Step 4: Verify key properties. Edge lengths: $|AB| = |BC| = |CD| = |DA| = 2$ (square base); $|EA| = \sqrt{0+4+9} = \sqrt{13}$, $|EC| = \sqrt{4+0+9} = \sqrt{13}$, $|ED| = \sqrt{0+0+9} = 3$, $|EB| = \sqrt{4+4+9} = \sqrt{17}$ (confirming the pyramid structure). ∴ Answer: AC
Correct Answer: AC